1. Introduction to virtual engineering Óbuda University John von Neumann Faculty of Informatics Institute of Intelligent Engineering Systems Lecture 2. Description of shapes in model space László Horváth university professor http://nik.uni-obuda.hu/lhorvath/

2. Definition of shape by its boundary Basic groups of shapes to be described Problem of boundary representation of shape Topological and geometrical entities Shape independence of topology Topological consistency Geometry: creating a curve Geometry: creating a surface Contents László Horváth ÓU-JNFI-IIES http://nik. uni-obuda.hu/lhorvath/

3. Definition of shape by its boundary László Horváth ÓU-JNFI-IIES http://nik. uni-obuda.hu/lhorvath/

4. Linear Curved Free form Analitical Generated according to predefined rule F1 F2 G1 Complex surface Basic groups of shapes to be described László Horváth ÓU-JNFI-IIES http://nik. uni-obuda.hu/lhorvath/

5. F1 F2 F1 F2 G12 G1 G2 L1 L2 F1 Connections of surfaces at intersection curves are to be described. Method: Topology (Euler) Problem of boundary representation of shape László Horváth ÓU-JNFI-IIES http://nik. uni-obuda.hu/lhorvath/

6. V E F V = vertex L = loop, ring E = edge, P = point G12 C = curve F = face S = Surface coedge Shell Consistent (complete) Shell + material = body Topological and geometrical entities (1) László Horváth ÓU-JNFI-IIES http://nik. uni-obuda.hu/lhorvath/

7. Prism – box = four prismatic segment Combination of solidsTopology Body = four lumps Topological and geometrical entities (2) László Horváth ÓU-JNFI-IIES http://nik. uni-obuda.hu/lhorvath/

8. Same topology for three different shapes Same structure Shape independence of topology László Horváth ÓU-JNFI-IIES http://nik. uni-obuda.hu/lhorvath/

9. Euler rule Leonhard Euler (1707-1783) swiss mathematican. Euler number for boundary of body: V - E + F Euler number is a constant V - E + F = C. For simple bodies ( no through holes or separated bodies (lumps) V - E + F = 2 Topological consistency Complete topology. Check by using of topological rules. Three or more edges must run into a vertex. Face must be enclosed by a closed chain of edges. Edge is included always in two loops for adjacent faces. Topological consistency (1) László Horváth ÓU-JNFI-IIES http://nik. uni-obuda.hu/lhorvath/

10. V-E+F=8-12+6=2 V-E+F=10-15+7=2 V-E+F=2-3+3=2 Topological consistency (2) László Horváth ÓU-JNFI-IIES http://nik. uni-obuda.hu/lhorvath/

11. TaskMethod Through specified pointsInterpolation Controlled by specified pointsApproximation P 0 P 1 P 2 P 3 According to specified rule Analitical Geometry: creating a curve László Horváth ÓU-JNFI-IIES http://nik. uni-obuda.hu/lhorvath/

12. Contour Generator Meridian curve Axis Direction of rotation Extension angle= 360 o Tabulated surface Rotational surface Geometry: creating a surface (1) László Horváth ÓU-JNFI-IIES http://nik. uni-obuda.hu/lhorvath/

13. Generator curve Path curve Spine Joint Profil curves Boundary curves Control of shape at the creation of a swept surface Geometry: creating a surface (2) László Horváth ÓU-JNFI-IIES http://nik. uni-obuda.hu/lhorvath/